WO1999019971A1

WO1999019971A1 - Three-degrees-of-freedom motor

Info

Publication number: WO1999019971A1
Application number: PCT/GB1998/003070
Authority: WO
Inventors: David Howe; Jiabin Wang; Geraint Jewell
Original assignee: The University Of Sheffield
Priority date: 1997-10-15
Filing date: 1998-10-14
Publication date: 1999-04-22
Also published as: GB2330457A; GB9721805D0; DE69810148D1; GB2330457B; EP1023762B1; ATE229705T1; EP1023762A1

Abstract

A three-degrees-of-freedom motor (10) comprises a stator (18) having four sets of coil pairs (A-A', B-B', C-C', D-D') distributed around the surface of a sphere and a spherically mounted rotor (12) having a payload arm (14) extending through an aperture (22) in the stator (18). The rotor (12) comprises a spherical permanent magnet comprised of four segments (26) parallel magnetised along lines parallel to the radius of curvature of each segment which bisects each segment so that the rotor has four magnetic poles.

Description

THREE-DEGREES-OF-FREEDOM MOTOR

The present invention relates to a motor having three degrees of freedom of movement, and to various devices including such a motor.

There is a growing need for programmable servo- controlled high-speed actuation in multiple axes, for applications as diverse as robotic manipulators, infrared and laser tracking systems, automated manufacturing, remote camera manipulation, and joysticks with force feedback. At present, multi-degree of freedom motions are realised almost exclusively by using a separate motor/actuator for each axis, which results in a relatively complicated and heavy transmission system. This inevitably compromises the dynamic performance and servo-tracking accuracy of the arrangement, due to the combined effects of inertia, backlash, non-linear friction, and elastic deformation of gears. Actuators with multiple degrees of freedom should alleviate these problems, whilst being lighter and more efficient. However, although they have been the subject of research for several decades, and numerous concepts have been proposed (see references [1] - [4]), their application potential has not been realised, probably due to the complexity of their structure and related difficulties in modelling their electro-magnetic behaviour and optimising their design.

A spherical actuator which is capable of two degrees of freedom has been described, systematically analysed and experimentally demonstrated in references [5] and [6]. These last two references describe a motor comprising a permanent magnet spherical rotor mounted for rotation about two orthogonal axis in a stator comprising coils adapted to induce magnetic fields through the rotor to rotate the rotor. The rotor comprises a diametrically parallel-magnetised, two-pole magnet, and the stator coils are arranged in three pairs of interconnected coils in a cubic arrangement.

The rotor has a payload arm which extends through an aperture of the stator and on which a payload can be mounted. One application of such an actuator is the manipulation of a camera. While two degrees of freedom motors such as described in references [5] and [6] can point a camera in any desired direction, it cannot rotate the camera to ensure the camera is aligned with an horizon. Therefore there is a need for a motor which not only enables pan and tilt of a rotor through about ±45°, but also which permits rotation of the rotor about its axis. However, such a motor should be capable of modelling of its magnetic behaviour to enable effective and precise control of its movement and also optimisation of its design to meet different design criteria.

In accordance with the present invention, therefore, there is provided a motor comprising a permanent magnet spherical rotor mounted for rotation about three orthogonal axes in a stator comprising coils adapted to induce magnetic fields through the rotor to rotate the rotor, characterised in that said rotor comprises a composite magnet having at least two pole pairs.

The stator preferably comprises four sets of coils, each set preferably arranged as a pair of two electrically interconnected coils.

The stator preferably has an access aperture and the rotor has a payload arm extending through the aperture, which aperture permits substantially ±45° pan and tilt rotation of the rotor arm and unrestricted rotation of the rotor arm about its own axis.

The rotor preferably comprises at least four segments of a sphere, each segment being magnetised so that the curved surface of the segment has substantially only one polarity.

The segments are preferably parallel magnetised, and, in relation to that radius of curvature which bisects each segment both transversely and longitudinally of the segment, the magnetisation lies parallel to that radius.

Preferably the stator comprises four rim coils arranged with their centres lying substantially on a rim plane of the rotor parallel said aperture, and four base coils arranged with their centres lying on a base plane, parallel said rim plane and disposed remote from said aperture, opposite pairs of said rim and base coils being electrically interconnected.

Preferably the base coils are rotationally offset with respect to the rim coils. Preferably there are only four segments and/or only four sets of coils.

Preferably the stator is surrounded by a spherical ferro-magnetic shell so as to increase magnetic coupling between the respective fields.

The rotor may similarly have a hollow centre or an iron core, the segments being shaped accordingly.

The invention also provides a robotic manipulator, an infrared or laser tracking system, a remote camera manipulation device, or a joystick with force feedback, comprising a motor as defined above.

The invention is further described hereinafter, by way of example, with reference to the accompanying drawings, in which :-

Figures la and lb are perspective schematic views of a three degree of freedom spherical motor in accordance with the present invention, Figure la being a perspective view from one side, while Figure lb is an underneath view of the motor; Figures 2a and b are schematic representations of the rotor of the motor of Figure 1, showing the magnetisation pattern thereof;

Figure 3 is a schematic illustration of the coil windings of the motor of Figure 1;

Figure 4a shows the spherical co-ordinate system; while Figure 4b is a plot of torque against motor size ratios;

Figure 5 illustrates the winding distribution of one coil;

Figure 6 is a sensor arrangement for a two-degrees- of-freedom motor;

Figure 7 is a block diagram of a suitable closed- loop control system for a two-degrees-of-freedom motor; Figure 8 is a schematic perspective view of the stator housing corresponding to Figure la;

Figure 9 is a view corresponding to Figure lb of the stator housing shown in Figure 8;

Figure 10 is a view in the direction of arrow X in Figure 9; and

Figures 11a to d are different views of one-half of a stator housing, Figure 11a being a plan view, Figure lib being a section on the line C-C in Figure 11a, Figure lie being a section on the line D-D in Figure 11a, Figure lid is a view in the direction of arrow D in Figure 11a and Figure lie is a view in the direction of arrow E in Figure 11a.

In the drawings, a spherical three-degrees-of- freedom of movement motor 10 is shown. The motor comprises a spherical rotor 12 having a payload arm 14. A stator 16 comprises a housing 18 and stator coils 20. The housing 18 has an aperture 22 through with the payload arm 14 extends. The motor 10 is adapted to move the payload arm through about ±45° panoramic movement in the x-z plane (indicated by the co-ordinate reference system 24 shown in Figure la) , and a similar ±45° tilt movement in the y-z plane. In addition, the motor is arranged to rotate the payload arm 14 about its own axis .

Eight stator coils 20 are distributed over the surface of the stator 18, as seen in Figure lb.

In Figure 2, the rotor 12 comprises four quarter spheres 26 bonded together to form the spherical rotor 12. The quarter spheres are parallel magnetised as indicated by the arrows 28 in Figure 2a so as to produce a rotor having four magnetic poles (indicated N and S) distributed around its circumference. The arrows 28 are parallel the radius of curvature of each segment, which radius bisects each segment. The straight edges 29 of each segment meet along a central line of the rotor.

Although not essential, it is preferred that the plane of the four magnetic poles (ie the plane of the rotor perpendicular to the straight edges 29) is arranged to lie in the plane containing the x axis as illustrated by the position of the payload arm in Figure 2. The composition of the rotor is not critical, but it may comprise NdFeB permanent magnet components bonded together. Moreover, the rotor may be hollow, solid or have an inner spherical iron core (not shown) .

Turning to Figure 3, as mentioned above there are eight stator coils 20 which are arranged in pairs, A-A', B-B', C-C and D-D'. Rim coils A, B, C and D are spaced around the periphery of the stator body 18 adjacent the aperture 22. Their centres are disposed on a rim plane, approximately equatorial with respect to the centre of rotation of the rotor 12. Base coils A', B', C and D' have their centres likewise disposed on a base plane of the stator 18, parallel to the rim plane and aperture 22, although remote from the aperture with respect to the rim coils A, B, C and D. Moreover, the base coils A', B', C and D' are rotationally offset with respect to the rim coils A, B, C and D so that the axis of the magnetic field established by any one coil pair passes through the rotor 12.

With reference to Figures 8 to 11, the stator body 18 comprises two halves, 18a, 18b having meeting faces 30 provided with connecting tags 32 by means of which the two halves, 18a, 18b, can be secured together.

Each coil 20 comprises a conical indentation 34 in the stator body 18 in order to receive a correspondingly shaped coil of wire (not shown) . At the base of each indentation 34 is a spigot 36 adapted to receive a screw fastening, or the like to secure a coil (not shown) in position. Of the rim coils A, B, C, D, the indentation 34 for the coils A and C are formed completely in one stator body 18 a or b; whereas the indentations 34 for the coils B, D are formed between the facing surfaces 30 of adjacent stator halves A, B. On the other hand, the base coil indentations 34 for the coils A', B', C and D' are formed two in each stator half 18a, b, as shown in Figure 9.

Although only two are shown, various bores 38 may be formed in the stator housing 18 for the insertion of Hall effect or like sensors (not shown) for detecting the position of the rotor 12 within the stator and providing feedback control.

The rotor 12 is received in a spherical surface 40 formed between the two halves, 18a, b of the stator. The stator body is moulded from a plastics material having a low friction surface so that the rotor is free to rotate within the confines of the stator without undue frictional resistance.

Thus by appropriate control of the excitation of the coil pairs in A-A', B-B ' , C-C and D-D' the rotor can be urged to rotate about any of the x, y or z axes or some combined complex movement. Indeed, it is an important aspect of the present invention that the characteristics of the motor be successfully analysed so that appropriate control functions can be developed to maximise the motor's efficiency. Thus if torque is the prime requirement, then torque can be maximised by appropriate dimensioning of the various components according to the mathematical analysis of the motor system. Because the coil and magnet system is integrated and yet still produces three degrees of freedom of movement of the rotor relative to the stator, the complexity and hence the cost of the motor is minimised. Moreover, because of the compactness of the design, relatively high power to weight is achieved for a direct drive system. The in-built position sensing enables precise feedback control so that precise holding of position and the load can be achieved.

Indeed, since it is fundamental to the practicality of the motor of the present invention that it be capable of control, knowledge of the magnetic field distribution produced by the spherical permanent magnet rotor is essential to establish an accurate model of the motor/ actuator 10, for design optimisation and dynamic modelling.

Without loss of generality, an air-cored actuator 10 is considered. Thus, the entire magnetic field region can be divided into two sub-regions, the outer airspace/ winding region in which the permeability is μo

and the magnet region in which the permeability is μo,μ_r- Therefore: in the airspace/windings

B in the magnet (1)

where μ_r is the relative recoil permeability of the magnet and M is the remanent magnetisation. For a permanent magnet having a linear demagnetisation characteristic, μ_r is constant and M is related to the

remanence, B_rem, by M = B_rem/μO . It is convenient to formulate the field distribution in terms of a scalar potential, φ defined as H--Vφ, and the spherical coordinate system shown in Figure 4a. This leads to the following field equations: f ₇2 _ Λ in the airspace/windings [V ² φ _π = V - M / μ _r in the magnet (2) The components of the magnetisation vector M shown in Figure 2a may be expressed as:

(3)

where sgn(-) denotes the sign function and

M_c=B_rem/μo^2 . It can be shown that V-M≡O . However, the contribution of the magnets will appear in the following interface boundary conditions:

where R_m is the radius of the spherical rotor. The radial component, M_r may be expanded into spherical harmonics of the following form:

M M lm P_t ^m (cos θ ) s in α (5)

where Pι^m(-) denotes the associated Legendre polynomial of degree 1 and order m, and M_lm is given by:

Solving for equation (2) with the boundary conditions of equation (4) yields the following expressions for the flux density distribution in the airspace/winding region: ^B _fr=

B w =

B la r^"(/+2)P_/ ^m(cos6»)cosQ;

where C_lm is given by:

)l

The torque exerted on the rotor, resulting from the interaction between the current in a stator winding and the rotor magnetic field, is given by:

T = - r x (J x B )d V ₍₈₎

where J denotes the current density vector in the winding region V. Each winding comprises a number of circular turns distributed on the spherical stator, and occupying an area bounded by r=R₀,r=R_s, and δ=δo_r δ=δ_±, as shown in Figure 5. Considering a single turn with infinitesimal cross-section ds = rdrdδ and an enclosing circular contour C, the total torque produced by the winding may be obtained from the following integration: r (9)

It can be shown that B_lr is dominated by its fundamental component, viz. 1=2, m=± . Therefore, the result of equation (9) neglecting high order harmonics, is given by:

where v = [v_xv_yv_z]^τ is the direction cosines of the winding axis, and the torque magnitude, T_m, is given by:

As can be seen, T_m is dependent upon B_rem and J as well as the geometrical parameters of the rotor and the winding. If the airgap between the rotor and winding is G, then for given B_rem J, δ₀, δι and R_{S f} T_m is a function of R„/R_s - This relation is plotted in Figure 4b assuming B_rem = 1-2T, μ_r = 1.15, J = 2.0A/mm², δ₀ = 5°, , = 30°, G = 0.002m and R_s = 0.042m. It is evident that there exists an optimal ratio of R_m/R_s viz. 0.738, that yields maximum torque .

The torque, T_c produced by a pair of windings

carrying current i and having direction cosines [v_xyv_z]^τ

and [v'_xV'yV'_zf , respectively, is given by:

T =

(11)

where

K_τ^T fl[J{Fξ - ^δ -δ^ (12) is defined as the winding torque constant, and N is the number of turns for the winding pair. The total torque, T_em, produced by four sets of identical winding, A, B, C,

and D, having direction cosines fv_XJv_yJv_ZJ]^τ and [v^f _XJ v' _yj v' _ZJ]^T (j = A, ...,D) is, therefore given by:

where i = [i_AiBiciD]^Υ is the winding current vector, and K_M, defined as the torque matrix of the actuator, is given by:

'y²A -^vz²A ⁺V²y_A-V]_A ^V]B ^{~ v}lβ + ^V']B zB

KTM - Ki ~^vxA^vyA ~ ^V'XA

~^vxB^v B - ^v'xB yB ^vxA 'zA + ^v'xA zA "xB ^vzB + V xB ^v zB

(14) ^v]c ~ ^vc + v'²c-v c Zla - v)_D + v'l_D-v']_D

~^vxC^VyC - ^v'xC ^v'yC ~^VxD^vyD ~ ^v' xD ^v' yD ^vxC^VzC ^{+ v}'xC^V'zC ^vxD^vzD ^{+ v}'xD ^v'zD

It can be shown that the rank of K-m is three within the working envelope of the actuator, implying that the actuator is able to deliver motion in three degrees-of- freedom, viz. ±45° pan-tilt excursions and continuous rotation.

Three degrees of freedom would also be possible with only three sets of coils, instead of the four described herein. However, there may then be a risk that singularities may be encountered and perhaps only a narrower working envelope would be possible. On the other hand, five or more sets of coils are also possible, except that the overall specific torque of the motor with increasing number of coils will decrease. Therefore the described number of four coil sets is preferred. A similar discussion may be followed in relation to the number of pole pairs of the rotor magnet, although two, in this case is not only the minimum but is also the preferred number. With greater number of poles the magnetic fields tend to interact and cancel one another.

By a similar integration procedure, the flux- linkage of a pair of windings due to the rotor magnetic field is given by:

¥ _ww =^~ K -"^* _EE ( v^v y_v v^v _zz +' V" y V ^r z J (15)

where K_E, defined as the back-emf constant of the winding pair, is identical to K_τ. The back-emfs of four identical windings A, B, C, and D, therefore, given by:

d_ ^e _i = -^Kτ τλv_yjv_zj + V_yj V_zj ) j = A ,...,D ₍₁₆₎ dt

For close loop control, it is necessary to have position feedback signals. In reference [6], which relates to a two-degrees-of-freedom motor, this is achieved by using two sensors, for example Hall effect or magnetoresistive sensors. However, in reference [7], referring to a similar two degree of freedom motor, the orientation of the rotor is represented by a set of Euler angles β, using four sensors positioned as shown in Figure 6. Sensors 1 and 2 are deployed symmetrically with respect to the x axis on the x-y plane, and sensors 3 and 4 are similarly deployed on the x-z plane. The magnetic field which is measured by the sensors is the combination of the radial components produced by all the field sources, viz., the rotor magnet and the winding currents, i_A, i_B and i_c, and the sensor outputs, V_SJ j = 1, 2,..., 4), may be written as:

V_SJ = k(B_mrj+B_Arj+B_Brj+B_Crj) j = 1,2,...,4 (17)

where k is the sensitivity, and B_{mrj /} B_Arj, B_Brj and B_Crj are the radial components of flux density at the sensor positions contributed by the rotor magnet and the currents in the windings A, B and C respectively. Due to the symmetry of the field and sensor system, equation (20) can be reduced to:

V_SJ = kB_mrj+k_sAi_A+(-i k_sBi_B j = 1,2

V_SJ = kB_mη+kJ_A+(-iyk_sCi_c j = 3,4 (18) where k_sA, k_sB and k_sc are coefficients relating the sensor outputs to the winding currents i_{A /} i_B and i_c. It is important to note that these coefficients are constant, and may be determined by either field computation or calibration. The component of output voltage, V_smj, contributed by the rotor magnet is given by:

V_smj = kB_r0 cosθ_SJ (19)

where B_r0 is a constant related to the rotor magnetic field distribution and Θ_SJ is the angle between the X_b axis (rotor magnetisation axis) and the j^th sensor position vector. If the rotor is displaced, from the initial orientation, by rotating it about the y_b axis by an angle β and about the z_b axis by an angle α, the

resulting cos#_SJ is given by:

cosθ_SJ = cosa₀cβca + (-l/sin gsa j⁼l,2

cosθ_SJ = cosa₀cβca + (-l)^!~!sina₀sβca j=3,4 (20) Combining equations (18) -(20) and solving for a and β

yields :

a= sm Vsl - V* ~ ^2ksB ;β= sin -ι l *-ksC C

2kB_r0 sin _Q 2&5_r0 sin<z₀ cosα (21)

In equation (21) , the sensor outputs are utilised differentially, which improves the noise rejection property of the position sensing system. The fact that

and β are dependent upon kB_c0sin requires some attention since, in general, both k and B^,, which is proportional to the remanence of the rotor magnet, will vary if the ambient temperature changes. However, this problem may be circumvented by performing an auto-calibration procedure prior to normal operation.

A complete dynamic model for the two-degrees-of- freedom actuator of reference [6] is given by:

^rMQ_E + CQ_E + G + τ J_/EE =K_Eτi_w

Li +Ri_w -K_E ^τ _TQ_E = u_E (22) where Qε - [βα]^τ are the Euler angles representing

the rotor orientation, the inertia matrix M, the Coriolis and centripetal force matrix C, and the gravitational torque vector G being given by:

where a shorthand notation for sine and cosine functions is used for clarity, that is, sα represents

sinα. I_b is the combined moment of inertia of the rotor and payload referred in the rotor co-ordinate system, _ε = [ u_Au_Bu_c]^τ is the winding terminal voltage vector, i_w = [i_Ai_Bi_c] is the winding current vector, L = diag [L_AL_BLc] ^τ is the diagonal winding self-inductance matrix, R = diag

[K_AΛ_βic]^τ is the diagonal winding resistance matrix, T^B is the vector representing the Coulomb and viscous friction, and K_Eτ_r defined as the actuator torque matrix, is related to the actuator torque constant K_τ by:

Note that equation (22) has a singularity at a =

90°. However, with the actuator design in reference [6]

the angular excursion of a is within ±45°, and this singular point will never be encountered. Also, it will be noted that in non-singular regions, equation (22) constitutes a Hamiltonian system, and, therefore, possesses a well-understood structure and similar important properties as the dynamic equations for robotic manipulators (reference [8] ) . As a result, any advanced control law for the control of robotic manipulators can be applied to the spherical actuator.

As an example, a robust outer PD position control law as described in reference [9] in conjunction with an inner PI current control law, as shown in Figure 7, is utilised for the control of the spherical actuator. The role of the inner current tracking loop is to minimise the effects of the back-emnf and current transients on the outer position servo loop, so that a robust design philosophy, espoused in reference [9], can be used to determine the control gain matrices K_v and A. The output of the position controller is two independent torque demands, but equation (22) has three independent control inputs. The extra degree of freedom in the control variables suggests that there exists a redundant control input which may be used for optimal control, eg to minimise the total energy consumption for a given torque demand. This control strategy is implemented by taking the weighted pseudo inverse of K_Eτ, as denoted by K^* _Eτ in Figure 7.

With the three-degrees-of-freedom motor of the present invention, a similar position sensing and control regime is within the ambit of the man skilled in the art except that further sensors will be required to detect the rotational position of the rotor, in addition to its position in relation to pan and tilt angles.

REFERENCES

1. K Davey, G Vachtsevanos and R Powers "The analysis of fields and torques in spherical induction motors" IEEE Trans, on Magn vol. 23, pp 273-282, 1987.

2. K Lee and C Kwan "Design concept development of a spherical stepper for robotic applications" IEEE Trans on Robot Automa t vol 7 pp 175-181, 1991. 3. B Bederson, R Wallace and E Schwartz "A miniature pan-tilt actuator: the spherical pointing motor" IEEE Trans on Robot Automat vol 10 pp 298-308, 1994.

4. K Lee, J Pei and R Roth "Kinematic analysis of a three degree of freedom spherical wrist actuator", Mechatronics, Vol 4 pp 581-605, 1994.

5. J Wang, G W Jewell and D Howe "Analysis, design and control of a novel spherical permanent magnet actuator" IEEE Proc Electr Power Appl in press, 1997.

6. J Wang, G W Jewell and D Howe "Development of a novel spherical permanent magnet actuator" Proc of IEEE Inter conf on Robotics and Automation New Mexico, USA 1190-1195, 1997.

7. J Wang, G W Jewell and D Howe "Development of a novel spherical permanent magnet actuator" IEEE

Colloquium on new topologies for PM machines London June 1997 pp 8.1-10.

8. F L Lewis, C T Abdallah and D M Dawson "Control of robotic manipulators", Mac illan Publishing Company, 1993. 9. J Wang, S J Dodds and W N Bailey "Guaranteed rates of convergence of a class of PD controllers for trajectory tracking problems of robotic manipulators with dynamic uncertainties", IEE Proc. -Control Theory Appl . , 1996 143(2), pp 186-190

GLOSSARY

List of principal symbols used in this specification

B = magnetic flux density, T

^Br m = remanence, T

G = airgap length, m

H = magnetic field strength, A/m i,, = winding current vector, A

J_w = winding current density, A/m²

K_τ = torque constant, N-m/A

K_E = back-emf constant, V-s/rad

L = inductance matrix, H

M₀ = remanent magnetisation, A/m

•R = resistance matrix, Ω

R_s = stator radius,

R_m = rotor radius, m s = Laplace operator, s^"1

T, τ = torque, N-m u_B = winding terminal voltage vector, V a,β, Φ = Euler angles, rad φ = scalar magnetic potential, A

Ψ_w - winding flux-linkage, Wb μ₀ = permeability of free space, H/m μ_r = relative recoil permeability = angular velocity, rad/s

Key to symbols used in Drawings

WA = inding Axis

RM = Rotor Magnet

A,B,C = Winding A,B,C

RMA = Rotor Magnetization Axis

SI...S4 = Sensor 1...4

PDPC = PD Position Controller

PICC = PI Current Controller

Bemf = Back emf

ET = effective Torque

FT = Friction Torque

AM = Actuator Model

Claims

1. A motor comprising a permanent magnet spherical rotor mounted for rotation about three orthogonal axes in a stator comprising coils adapted to induce magnetic fields through the rotor to rotate the rotor, characterised in that said rotor comprises a composite magnet having at least two pole pairs.

2. A motor as claimed in claim 1, wherein the stator comprises four sets of coils, each set preferably arranged as a pair of two electrically interconnected coils.

3. A motor as claimed in any preceding claim, wherein the stator has an access aperture and the rotor has a payload arm extending through the aperture, which aperture permits substantially ┬▒45┬░ pan and tilt rotation of the rotor arm and unrestricted rotation of the rotor arm about its own axis .

4. A motor as claimed in claim 2 and 3, wherein the centrally positioned rotation axis is the axis of the rotor arm when centrally located in said access aperture.

5. A motor as claimed in any preceding claim, wherein the said rotor comprises at least four segments of a sphere, each segment being magnetised so that the curved surface of the segment has substantially only one polarity.

6. A motor as claimed in claim 5, wherein the segments are parallel magnetised, and, in relation to that radius of curvature which bisects each segment both transversely and longitudinally of the segment, the magnetisation lies parallel to that radius.

7. A motor as claimed in claim 5 or 6, when dependent on claim 3, wherein the straight edges of adjacent segments are coincident with the rotor arm axis .

8. A motor as claimed in claim 7, wherein there are four segments only, each a quarter sphere.

9. A motor as claimed in any preceding claim, wherein the stator coils comprise at least four rim coils arranged with their centres lying substantially on a rim plane parallel said aperture, and a corresponding number of base coils arranged with their centres on a base plane, parallel said rim plane and disposed remote from said aperture, opposing pairs of said rim and base coils being electrically interconnected.

10. A motor as claimed in claim 9, wherein the base coils are rotationally offset with respect to the rim coils.

11. A motor as claimed in claim 9 or 10, wherein there are four sets of coils only.

12. A motor as claimed in any preceding claim, wherein the stator is surrounded by a spherical, ferro-magnetic shell, so as to increase magnetic coupling between the respective fields.

13. A motor as claimed in claim 5 or in any of claims 6 to 12, when dependent on claim 5, wherein the rotor has a hollow centre or an iron core, the segments being shaped accordingly.

14. A motor comprising a permanent magnet spherical rotor mounted for rotation about three orthogonal axes in a stator comprising coils adapted to induce magnetic fields through the rotor to rotate the rotor, characterised in that the flux density created by the rotor in the regions of the coils is, in a spherical coordinate system, given by the following expressions :

Γê₧ / B _Ir = V C_;m(/ + l)r^"('^{+ )}?,^ra(cos^)sin╬▒

7=2,4,... Γäó = 1,3,...

where C_lm is given by:

C_lm =- ╬╝₀R_m ^l+2M_lm / [(l ₊ l(l + ╬╝_r)]

15. A motor as claimed in claim 14, wherein the torque magnitude of the motor is optimised by arranging the coils so that the following expression is true: T_m

ln(╬¢, 1 7?₀)(sin³ ╬┤_x -sin³ ╬┤₀) / Z(2╬╝_r +3)

16. A robotic manipulator, an infrared or laser tracking system, a remote camera manipulation device, or a joystick with force feedback, comprising a motor as claimed in any preceding claim.

17. A motor substantially as hereinbefore described with reference to the accompanying drawings .